Problem: Simplify the following expression: $y = \dfrac{4x^2+17x- 15}{x + 5}$
First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(4)}{(-15)} &=& -60 \\ {a} + {b} &=& &=& {17} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-60$ and add them together. Remember, since $-60$ is negative, one of the factors must be negative. The factors that add up to ${17}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-3}$ and ${b}$ is ${20}$ $ \begin{eqnarray} {ab} &=& ({-3})({20}) &=& -60 \\ {a} + {b} &=& {-3} + {20} &=& 17 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({4}x^2 {-3}x) + ({20}x {-15}) $ Factor out the common factors: $ x(4x - 3) + 5(4x - 3)$ Now factor out $(4x - 3)$ $ (4x - 3)(x + 5)$ The original expression can therefore be written: $ \dfrac{(4x - 3)(x + 5)}{x + 5}$ We are dividing by $x + 5$ , so $x + 5 \neq 0$ Therefore, $x \neq -5$ This leaves us with $4x - 3; x \neq -5$.